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Syllabus
3rd Year Honours Options for Arts MA360.
CS304 Mathematical and Logical Aspects of Computing
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An appreciation of some of the mathematical and logical ideas and techniques
which are useful in computer science.
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Boolean Logic, Path searching logic trees.
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Simple machines; Finite State Machines; Turing Machines.
MA385 Numerical Analysis
[There is a practical assessment in computing, carrying
up to 30% of the total marks.]
Modelling with first order differential equations.
Euler's method and convergence.
Higher order one-step methods: Heun's method, modified Euler method, Runge-Kutta
method of order 4.
Practical implementation of the Runge-Kutta method using variable step
size.
Introduction to C programming on the VAX.
Round-off errors: machine operations, well-conditioned problems, numerical
trustworthiness of algorithms, numerical stability of algorithms, error
estimation.
Round-off errors and one-step methods.
Shooting method for higher order (non-linear) differential equations.
Iterative methods for finding zeros of non-linear functions: convergence,
Aitken's accelerated convergence, quadratic convergence and Newton's method.
Computer implementation of the shooting method for a non-linear second
order boundary value problem.
Outline of the finite element method for a two-dimensional Dirichlet boundary
value problem.
Text:
J. Stoer & R. Bulirsch. Introduction to Numerical Analysis.
Springer.
MA378 Numerical Analysis
Numerical quadrature in one variable (and polynomial interpolation): Monte
Carlo integration, Newton-Cotes integration and associated errors.
Gaussian integration (existence, uniqueness and errors of interpolating
polynomials, Neville's algorithm, divided differences, Gram-Schmidt process
and orthogonal polynomials).
Numerical quadrature in several variables.
Systems of linear equations: Gaussian elimination.
Finer points on the finite element method: completion of a matrix space,
dependence of errors on triangulations.
Text:
J. Stoer & R. Bulirsch. Introduction to Numerical Analysis.
Springer.
CS402 Cryptography
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Number theory: time estimates, finite fields and quadratic residues
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Cryptography: public key cryptography, RSA cryptosystems, the Diffie-Hellman
(discrete log) key exchange system, knapsack method
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Primality and factoring: the Rho method, factor bases, the continued fraction
method
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Elliptic curves: elliptic curve cryptosystems, elliptic curve factorisation
Texts:
N.Koblitz, "A Course in Number Theory and Cryptography" (Springer)
MA410 Artificial Intelligence
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Review of logic
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Propositional calculus, truth tables, conjunctive and disjunctive normal
forms, arguments
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Predicate calculus, Skolemisation, clause form, resolution
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Searching: Breadth first, depth first and best first search in a state
space
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Two-person games, the minimax procedure, alpha beta pruning
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Introduction to PROLOG
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Facts, rules, queries, back-tracking, lists
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Searching, production systems
References
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I. Bratko, "PROLOG programming for artificial intelligence" (Addison Wesley)
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G.F. Luger and W.A. Stubblefield, "Artificial intelligence and the design
of expert systems" (Benjamin Cummings)
MA416 Ring Theory
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Introductory examples of rings and fields
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Axioms. Subrings. Integral domains; theorems of Fermat and Euler
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Field of quotients of an integral domain
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Division rings. Quaternions.
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Rings of polynomials. Factorisation. Gauss's Lemma. Eisenstein's criterion
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Ideals, factor rings, ring homomorphisms. Homomorphism theorems
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Prime ideals, maximal ideals. Principal ideal rings
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Unique factorsation domains, Euclidean domains.
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Gaussian integers
MA491 Field Theory
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Field extensions - simple, algebraic, transcendental
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The degree of an extension
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Ruler and compass constructions
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Algebraically closed fields, splitting fields and finite fields
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Galois groups and the Galois correspondence
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Introduction to codes (ISBN, linear, cyclic)
Text
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J.B. Fraleigh, "A First Course in Abstract Algebra" (Addison Wesley)
References
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I.T. Adamson, "Introduction to Field Theory" (Oliver & Boyd)
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I.N. Herstein, "Topics in Algebra" (Wiley)
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I.N. Stewart, "Galois Theory" (Chapman & Hall)
MA484/486 Statistics
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Introduction to statistical inference: methods of estimation, least squares
and maximum likelihood, Bayes methods, properties of estimators
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Approaches to statistics, including data analytic, frequentist, Bayesian,
robust, non-parametric, structural and fiducial. Ther role of probability
in the frequentist approach to statistical inference.
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Brief review of random variables and their distributions. Some methods
of point estimation needed in hypothesis tests, including maximum likelihood
and method of moments.
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Hypothesis testing: likelihood ratio tests, Neymann Pearson theory. Exponential
families of distributions. Derivation of uniformly most powerful tests
when they exist.
Discussion of uniformly most powerful unbiased tests. Discussion of
uniformly most accurate, and uniformly most accurate unbiased confidence
sets. Applications.
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Principles of data reduction, with particular emphasis on the concept of
sufficiency. The concept of completeness.
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Point estimation: criteria and derivations. Methods of estimation, especially
minimum variance unbiased estimators. Basu's Theorem. Applications.
Texts
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R. Hogg & A. Craig, "Introduction to Mathematical Statistics" (Macmillan)
References
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L.J. Bain & M. Engelhardt, "Introduction to Probability and Mathematical
Statistics" (van Nostrand Reinhold)
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E.J. Dudewicz & S.N. Mishra, "Modern Mathematical Statistics" (Wiley)
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P.G. Hoel, S.C. Port & C.J. Stone, "Introduction to Statistical Theory"
(Houghton Mifflin)
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H.J. Larson, "Introduction to the Theory of Statistics" (Wiley)
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A.M. Mood, F.A. Graybill & D.C. Boes, "Introduction to the Theory of
Statistics" (McGraw Hill)
MA490 Measure Theory
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The Lebesgue integral: the deficiencies of the Riemann integral, Lebesgue
measure, measurable functions, the Lebesgue integral
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Convergence theorems, functions of bounded variation and absolutely continuous
functions, Vitali's Covering Theorem, integration and differentiation
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General measure and integration theory: outer measures, measures, measurable
functions, modes of convergence
Texts
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H.L. Royden, "Real Analysis" (Collier Macmillan)
References
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R.G.Bartle, "The Elements of Integration" (Wiley)
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A. Brown & C. Pearcey, Introduction to Operator Theory, Vol I (Springer)
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E. Hewitt & K. Stromberg, "Real and Abstract Analysis" (Springer)
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M. Spiegel, "Real Variables" (Schaum Outline)
MA482 Functional Analysis
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Normed vector spaces and inner product spaces, bounded linear mappings,
linear functionals and the dual space
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The classical Banach spaces and their duals, Hilbert spaces, orthogonal
decomposition, orthonormal bases
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Fourier series
Texts
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E. Kreyszig, "Introductory Functional Analysis with Applications" (Wiley)
References
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W. Rudin, "Functional Analysis" (McGraw Hill)
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W. Rudin, "Real and Complex Analysis" (McGraw Hill)
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M. Schechter, "Principles of Functional Analysis" (Academic Press)
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N. Young, "An Introduction to Hilbert Spaces" (Cambridge)
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